For bounded scalar fields, both the numerical solutions and analytic approximations are used to solve for \(\omega \)  from Eq. (REF ). To simplify the numerical calculation, Leaver proposed a continued fraction method for massless fields [1]}, [2]}. The eigen-equation is converted into a three-term recurrence relation. Leaver's method was developed for massive scalar fields in Ref. [3]} and refined later in Ref. [4]}. On the other hand, the analytic approximation at the limit \(\mu M \ll 1\)  is obtained in Ref. [5]}. This approximation is not consistent with the numerical solution, causing doubts about the reliability of both results. The problem is recently resolved in Ref. [6]} by including the next-to-leading-order (NLO) contribution in the analytic result. At the other limit \(\mu M \gg 1\) , a JWKB estimate of the fastest growth rate is given in Ref. [7]}.
